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Estimating and localizing the algebraic and total numerical errors using flux reconstructions
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SYSNO ASEP 0481663 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Estimating and localizing the algebraic and total numerical errors using flux reconstructions Author(s) Papež, Jan (UIVT-O) RID, SAI
Strakoš, Z. (CZ)
Vohralík, M. (FR)Source Title Numerische Mathematik. - : Springer - ISSN 0029-599X
Roč. 138, č. 3 (2018), s. 681-721Number of pages 41 s. Language eng - English Country DE - Germany Keywords numerical solution of partial differential equations ; finite element method ; a posteriori error estimation ; algebraic error ; discretization error ; stopping criteria ; spatial distribution of the error Subject RIV BA - General Mathematics OECD category Applied mathematics R&D Projects GA13-06684S GA ČR - Czech Science Foundation (CSF) Institutional support UIVT-O - RVO:67985807 UT WOS 000426063200006 EID SCOPUS 85028846639 DOI https://doi.org/10.1007/s00211-017-0915-5 Annotation This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in H(div,omega), whereas the lower algebraic and total error bounds rely on locally constructed H01(omega)-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over the computational domain and illustrate the associated cost. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2019
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